The theory of powerful p-groups was created by A. Lubotzky and A. Mann [1987]; it was also anticipated in an earlier work of M. Lazard [1965]. Powerful p-groups have already found several applications in the theories of finite p-groups, of pro-p-groups, of residually finite groups, of groups with bounded ranks, of groups of given coclass, etc. One can say that the theory of powerful p-groups reflects the properties of the “linear part” of a finite p-group of given rank. Applications to finite p-groups with almost regular p-automorphisms are based on the bounds for the ranks in terms of the number of fixed points and the order of the automorphism (§2.2). The exposition in this chapter follows [A. Lubotzky and A. Mann, 1987] and includes some lemmas from [A. Shalev, 1993a] and [J. D. Dixon et al., 1991]. The proofs, however, are here inflated to a more verbose form, to make them accessible for a beginner; some sharper bounds are sacrificed for the same reasons. We shall consider only the case when p is an odd prime; the same results hold for p = 2, but the definitions and some proofs are a little different (although not more difficult) and are left as exercises to the reader.

Definitions and basic properties

Throughout the chapter, p denotes a fixed prime number, which is assumed odd, if not otherwise stated.

Now we are in a position to prove the first of the main theorems on finite p-groups with p-automorphisms having few fixed points. If such an automorphism is “almost regular”, with pm fixed points, then the group is “almost nilpotent”: it has a subgroup of (p, m)-bounded index and of nilpotency class at most h(p), where h is Higman's function. This bound for the nilpotency class of a subgroup of (p, m)-bounded index is best possible, if required to depend on the order of the automorphism only. The result of this chapter will be used in Chapters 13 and 14.

Theorem 8.1.If a finite p-group P admits an automorphism φ of prime order p with exactly pm fixed points, then P has a characteristic subgroup of (p,m)-bounded index which is nilpotent of class at most h(p), where h(p) is the value of Higman's function.

The proof relies on Higman's Theorem from § 7.2 on regular automorphisms of Lie rings in its combinatorial form and on the use of the associated Lie rings. Note that, at a first glance, an application of Higman's Theorem to L(P) and the induced automorphism φ cannot give us much information, since not only is φ not regular, but the number of fixed points of φ on L(P) can be much greater than on P, by a factor equal to the nilpotency class, say. Anotherimportant tool in the proof is a theorem of P. Hall from § 4.2.

In the extreme case, where a p-automorphism of a finite p-group has only P fixed points, the result is extremely strong.

Theorem 13.1.If a finite p-group P admits an automorphism φ of order pn with exactly P fixed points, then P has a subgroup of (p,n)-bounded index which is nilpotent of class at most 2 (abelian, if P = 2).

For |φ| = p this was proved by C. R. Leedham-Green and S. McKay [1976] and by R. Shepherd [1971]; in the general case it was proved by S. McKay [1987] and by I. Kiming [1988].

We give a proof which is different from the original ones; although with possibly worse bounds for the index of the subgroup, our proof is more Lie ring oriented, making use of Higman's and Kreknin's Theorems from Chapter 7, the theory of powerful p-groups from Chapter 11, and the Lazard Correspondence from Chapter 10. As in Chapters 8 and 12, bounds for the ranks of abelian sections allow us to assume P to be powerful. Using a generalization of Maschke's Theorem, one can show that every (φ-invariant abelian section is a kind of “almost one-dimensional” ℤ〈φ〉-module. This information is used in a reduction to the case where P is uniformly powerful, and later in the proof of a Lie ring theorem. An application of Higman's Theorem to a subring of the associated Lie ring L(P) allows us to assume P to be nilpotent of class h(p), the value of Higman's function.

This is a compilation of the lectures given in 1990–97 in the universities of Novosibirsk, Freiburg (in Breisgau), Trento and Cardiff. The book gives a concise account of several, mostly very recent, theorems on the structure of finite p-groups admitting p-automorphisms with few fixed points. The proofs, given in full detail, require various powerful methods of studying nilpotent p-groups; these methods are presented in the manner of a textbook, accessible for students with only a basic knowledge of linear algebra and group theory. Every chapter ends with exercises which vary from elementary checks to relevant results from research papers (but none of them is referred to in the proofs).

By the classical theorems of G. Higman, V. A. Kreknin and A. I. Kostrikin, a Lie ring is soluble (nilpotent) if it has a fixed-point-free automorphism of finite (prime) order. (These Lie ring theorems are also included along with all necessary preliminary material.) Prompted by and based on these Lie ring results, the main theorems of the book state that a finite p-group is close to being soluble (nilpotent) in terms of the order of a p-automorphism and the number of its fixed points. These results can be viewed as general structure theorems about finite p-groups. They are closely related to the theory of (pro-) p-groups of maximal class and given coclass and have natural extensions to locally finite p-groups.

Presenting linear (mostly Lie ring) methods in the theory of nilpotent groups is another main objective of the book.

The theorems of G.Higman, V. A.Kreknin and A. I. Kostrikin on regular automorphisms of Lie rings can be viewed as combinatorial facts about (Z/nZ)- graded Lie rings: they are actually proved as such, and it is in this form that they are used in studying p-automorphisms of nilpotent p-groups. We shall first prove Kreknin's Theorem for graded Lie rings using the varietal criterion from §5.2, which simplifies the proof to a few lines. (A longer version which gives an explicit upper bound for the derived length is indicated in the exercises.) Then nilpotency is derived from solubility in the case of the automorphism of prime order, again for graded Lie rings. Free Lie rings allow us to derive the required combinatorial consequences for arbitrary Lie rings. The theorems on Lie rings and finite nilpotent groups with regular automorphisms are also obtained as corollaries of these combinatorial facts.

Graded Lie rings

For the definition of graded Lie rings, see § 5.1. We begin with a version of a theorem of V. A. Kreknin [1963].

Theorem 7.1.Let n be a positive integer and suppose that L = L0 ⊕ L1 ⊕ … ⊕ Ln–1 is a (ℤ/nℤ)-graded Lie ring with components Ls satisfying [Li, Lj] ⊆ Li+j, where i + j is a residue mod n. If L0 = 0, then L is soluble of n-bounded derived length: L(k(n)) = 0 for some function k(n) depending only on n.

Throughout this chapter, p denotes a prime number. We prove here some elementary properties of finite p-groups including the Burnside Basis Theorem. Then we prove a theorem of P. Hall on the orders of the lower central factors of a normal subgroup. Many other properties of finite p-groups will be proved later, some in Chapter 6 using the associated Lie rings, some in Chapter 10 using the Mal'cev–Lazard correspondence, some in Chapter 11 on powerful p-groups. The main results of the book in Chapters 8, 12, 13 and 14 are also about finite p-groups.

We shall freely use the fact that the homomorphic images of commutator subgroups are commutator subgroups of the images (1.14), the same being true for verbal subgroups, like Gn = 〈gn | g ∈ G〉, by Lemma 1.47.

Basic properties

By the definition from § 1.1, a group is a p-group if the orders of all of its elements are powers of p. By Lagrange's Theorem, any group of order pn, n ∈ N, is a finite p-group. The converse is also true by the Sylow Theorems. Thus, we can safely redefine finite p-groups as groups of order pn, n ∈ N. By Lagrange's Theorem, all factor-groups and all subgroups of a finite p-group are again finite p-groups. Note that every group of order p is cyclic, since every non-trivial element generates a subgroup of order that divides p and hence equals p.

The second of the main results on almost regular p-automorphisms of finite p-groups is a match to Kreknin's Theorem on regular automorphisms of Lie rings. If a finite p-group P admits an automorphism φ of order pn with exactly pm fixed points, then P contains a subgroup of (p, m, n)-bounded index which is soluble of (p, n)-bounded derived length (that is, of derived length bounded in terms of the order of the automorphism only). Kreknin's Theorem is used twice in the proof. First it is applied to the associated Lie ring L(P), in the case where P is uniformly powerful, to prove that P is an extension of a group of (p, m, n)-bounded nilpotency class by a group of (p, n)-bounded derived length (this already gives a “weak” bound, in terms of p, m and n, for the derived length of P in the general case). Then free nilpotent ℚ-powered groups and the Mal'cev Correspondence are used to derive a consequence of Kreknin's Theorem, with a kind of a “weak” conclusion that depends on the nilpotency class. Rather miraculously, a combination of two “weak” results yields the desired “strong” bound, in terms of pn only, for the derived length of a subgroup of (p, m, n)-bounded index.

By Lemma 2.12 the number of fixed points of φ in all φ-invariant sections of P is at most pm; by Corollary 2.7 all these sections have rank at most mpn. This is why powerful p-groups appear naturally in the proofs.

Theorem 8.1 states that if a finite p-group P admits an automorphism of order P with pm fixed points, then P has a subgroup of (p, m)-bounded index which is nilpotent of p-bounded class. In this chapter we prove that the nilpotency class of a subgroup of (p, m)bounded index can be bounded in terms of m only. The following theorem is due to Yu. Medvedev [1994a,b].

Theorem 14.1.If a finite p-group P admits an automorphism φ of prime order p with exactly pm fixed points, then P has a subgroup of (p, m)-bounded index which is nilpotent of m-bounded class.

Neither Theorem 8.1 nor Theorem 14.1 follows from the other: if P is much less than m, then Theorem 8.1 gives a better result; on the other hand, if m is much less than p, then Theorem 14.1 is better. Theorem 14.1 confirmed the conjecture from [E. I. Khukhro, 1985] (also [Kourovka Notebook, 1986, Problem 10.68]). This conjecture was prompted by the result of C. R. Leedham-Green and S. McKay [1976] and R. Shepherd [1971] on p-groups of maximal class, which amounts to the special case of Theorem 13.1 where |φ| = |Cp(φ)| = P implies that P has a subgroup of p-bounded index which is nilpotent of class 2.

The proof of Theorem 14.1 is essentially about Lie rings; we use many of the of techniques developed in Chapter 13, including the lifted Lie products from [Yu. Medvedev, 1994b].

We construct both free nilpotent groups and free nilpotent Lie ℚ-algebras within associative ℚ-algebras. The Baker–Hausdorff Formula is proved to be a Lie polynomial which links the operations in the group and the Lie algebra. The construction is used to embed any torsion-free nilpotent group in its ℚ-powered “hull”. In Chapter 10, all this will be applied to establish the Mal'cev Correspondence between nilpotent ℚ-powered groups and nilpotent Lie ℚ-algebras and the Lazard Correspondence for nilpotent p-groups and Lie rings of class ≤ p – 1.

Free nilpotent groups

In § 5.3 we used a free associative ℚ-algebra A to construct a free Lie ring L as a subring of A(–) We use new “calligraphic” letters for these objects, since here we prefer to denote by A = A/Ac+1 and L = L/γc+1(L) the free nilpotent factor-algebras. In this section, we construct a free nilpotent group F within A with adjoined outer unity; A is the common ground for both L and F, which helps to establish connections between them.

We recall some definitions and basic properties. Let A be a free nilpotent associative ℚ-algebra of nilpotency class c with free (non-commuting) generators x1, x2, … (when necessary, we shall take a well-ordered set of generators of any given cardinality); “nilpotent of class c” means that every product of any c + 1 elements is 0.